Let $U$ be a smooth variety over $\mathbb{C}$. We know that there exists a smooth compactification $X$ such that $X-U$ is a normal crossings divisor $D$ and that the de Rham cohomology of $U$ can be computed using the complex of sheaves with logarithmic differentials on $X$: $$ H_{dR}^\ast(U)=\mathbb{H}^\ast(X, \Omega^\bullet_X(\log D)) $$

In general, one needs hypercohomology here but I am wondering if, when $U$ is affine, one simply has $$ H^\ast_{dR}(U)=H^\ast(\Gamma(\Omega^\bullet_X(\log D)) $$ I know that this is true if one uses instead the de Rham complex $\Omega^\bullet_U$ to compute cohomology and I know that $j_\ast \Omega_U^\bullet$ and $\Omega_X^\bullet(\log D)$ are quasi-isomorphic (here $j: U \hookrightarrow X$). Is this enough to conclude?